Higgs Picture of the QCD Vacuum

44 Pages·2004·0.374 MB·English

by  C. Wetterich

Higgs picture of the QCD-vacuum
C. Wetterich1
Institut fu¨r Theoretische Physik
Universit¨at Heidelberg
4 Philosophenweg 16, D-69120 Heidelberg
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2

t
c
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June 12, 2006
1
v
Abstract
7
5
0 The functional integral for QCD is reformulated by introducing
0 explicitly an integration over the fluctuations of composite quark-
1
antiquark bound states. Chiral symmetry breaking by the color sin-
4
0 glet scalar fieldinducesmassesforthefermions. Ourformulation with
/ scalar fluctuations may be useful for lattice gauge theories by modify-
h
p ing the spectrum of the Dirac operator in the vacuum and permitting
-
a simple connection to chiral perturbation theory. We propose that a
p
e “condensate” of quark-antiquark boundstates in thecolor octet chan-
h
nel generates masses of the gluons by the Higgs mechanism. A simple
:
v
effective action for quarks, gluons and (composite) scalars yields a
i
X surprisingly good description of the charges, masses and interactions
r of all low mass physical excitations - baryons, pseudoscalars and vec-
a
tor mesons. Dressed quarks appear as baryons and dressed gluons as
vector mesons.
1e-mail: [email protected]
1 Introduction
An analytic description of the vacuum in QCD remains a central goal in
quantum field theory. We have witnessed convincing progress of numerical
simulations of QCD on a lattice. Still, even an approximate analytic under-
standing would be a highly valuable complement. Computations of strong
cross sections and decay rates are very hardin lattice QCD. Simulations have
a long way to go before the properties of nuclei can be explained. Further-
more, a huge present and future experimental program tries to gather infor-
mation about the QCD phase transition and the phase diagram as function
of baryon density and temperature. Simulations for high baryon density are
notoriously difficult and an analytical understanding would be very helpful.
Akey issue forthesimplicity andsuccess ofananalytical description isan
efficient description of the relevant degrees of freedom. In QCD the relevant
degrees of freedom depend on the momentum scale. Processes involving high
momenta (more precisely high virtuality) are well described by quarks and
gluons. Perturbation theory based on the microscopic action for quarks and
gluons with a small gauge coupling yields reliable results. In contrast, the
relevant degrees of freedom at long distances or small momenta are mesons
and baryons. An efficient description of the vacuum should therefore involve
degrees of freedom for the low mass mesons. Typically, the lowest mass ex-
citations in the QCD vacuum should comprise the pseudoscalar meson octet
andsinglet(η ), thevector mesonoctetandthebaryonoctetintherespective
′
channels. We observe that this spectrum of real QCD differs strongly from
pure QCD (gluodynamics) without quarks where the low excitations consist
of glueballs. This simple fact suggests that an efficient analytic description
of real QCD at low momentum differs substantially from gluodynamics. As
the momentum scale is lowered an analytic description of real QCD should
effectively switch from gluons and quarks to mesons and baryons.
The most prominent long distance degrees of freedom are scalar quark-
antiquark (q¯q) bound states. The expectation value of such a composite
scalar field induces spontaneous chiral symmetry breaking. The excitations
of this scalar field describe then the associated (pseudo-) Goldstone bosons
π, K, η as well as the η . Our first task for an analytic description will there-
′
fore be to supplement quarks and gluons by the degrees of freedom for com-
posite scalar (and pseudoscalar) meson fields. In the first part of this work
(sect. 2) we will reformulate the functional integral for QCD in an exactly
equivalent form which comprises an integration over explicit scalar degrees of
freedom. Thesescalarsareofminorimportanceathighmomentabutbecome
crucial for the properties of the vacuum. Besides the analytical advantage of
a simple connection to chiral perturbationtheory this reformulation may also
1
offer important benefits for lattice simulations. The spectrum of the Dirac
operator acquires a mass gapeven in the chiral limit andthe contact to chiral
perturbation theory at long distances should become straightforward.
Concerning the QCD phase transition the task for an analytical descrip-
tion becomes even more involved: the formalism should now describe simul-
taneously quarks, gluons, mesons and baryons. Above the critical temper-
ature a quark gluon plasma is a valid approximation and explicit mesons
or baryons play no important role. In the hadron gas below the critical
temperature the mesons (and baryons) dominate whereas quarks and gluons
become irrelevant. A simple analytical description has to capture all these
degrees of freedom andprovide for a mechanism explaining why their relative
importance changes abruptly as a function of temperature.
Associating spontaneous chiral symmetry breaking with the phase transi-
tion yields a simple explanation why the quarks disappear from the relevant
spectrum below the critical temperature. T : the fermions get massive for
c
T < T . What is needed is a similar mechanism for the gluons. For the
c
electroweak phase transition the effective generation of a mass for the W-
and Z-bosons is well known. As the universe cools down below the critical
temperature both the gauge bosons and the fermions acquire a mass due to
the Higgs mechanism associated to the “spontaneous breaking” of the elec-
troweak gauge symmetry. The second part of this work will review a similar
“Higgs mechanism” for QCD, namely the “spontaneous breaking of color”
by the expectation value of composite q¯q-scalars in the color-octet channel
[1].
Strictlyspeaking, localsymmetries cannotbebrokenspontaneouslyinthe
vacuum. This has led to the realisation that the confinement and the Higgs
description are not necessarily associated to mutually exclusive phases. They
may only be different facets [3] of one and same physical state2 We stress
that this observation is not only of formal importance. For example, the
high temperature phase transition of electroweak interactions with a small
Higgs scalar mass ends for a larger scalar mass in a critical endpoint. Beyond
this endpoint the phase transition is replaced by an analytical crossover [4].
In this region – which is relevant for a realistic Higgs mass in the standard
model – a Higgs and a confinement description can be used simultaneously.
Our picture of the QCD vacuum resembles in many aspects the “strongly
coupled electroweak theory” at high temperature3. We will present a Higgs
2The complementarity between the Higgs and confinement description has been con-
sidered earlier for toy models with fundamental colored scalar fields [2].
3Withoutadirectconnectiontothe standardmodeltheSU(2)-Yang-Millstheorywith
strong gauge coupling and fundamental scalar has been first simulated on the lattice in
[5].
2
description of QCD as a complementary picture to the usual confinement
picture. Both the Higgs description and the confinement picture are consid-
ered as valid descriptions of one and the same physical properties of QCD.
A valid Higgs picture has to be consistent with the well established results
of the confinement picture and of lattice QCD.
With respect to color the composite scalar q¯q-bilinears transform as sin-
glets and octets, each in the (¯3,3)-representation of the SU(3) SU(3)
L R
×
chiral flavor symmetry. We will explore the hypothesis that the scalar octets
induce “spontaneous color symmetry breaking” while a “physical” global
SU(3) symmetry is preserved. This global symmetry can be used to classify
the spectrum of excitations according to the “eightfold way”. We find that
all gluons acquire a mass by the “Higgs mechanism” and belong to an octet
of the physical SU(3)-symmetry. Our picture provides an effective infrared
cutoff for real QCD by mass generation. (The cutoff for gluodynamics is
expected to be different!)
The Higgs mechanism also gives integer electric charge and strangeness
to all physical particles according to their SU(3)-transformation properties.
Furthermore, the expectation value of the quark-antiquark color octet breaks
the globalchiral symmetry. In consequence thefermions become massive and
the spectrum contains light pions and kaons as pseudo-Goldstone bosons. In
the limit of equal masses for the three light quarks the global vector-like
SU(3)-symmetry of the “eightfold way” becomes exact. The nine quarks
transform as an octet and a singlet. We identify the octet with the low mass
baryons – this is quark-baryon duality. The singlet is associated with the Λ
(1405) baryon and has parity opposite to the nucleons. Similarly, the eight
gluons carry the quantum numbers of the light vector mesons ρ,K ,ω. The
∗
identification of massive gluons with the vector mesons is called gluon-meson
duality.
Weproposethatthemainnonperturbativenewingredientfortheeffective
action of low-momentum QCD consists of scalar fields representing quark-
antiquark bound states. Once these composite operators are treated on the
same footing as the quark and gluon fields, the description of propagators
and vertices becomes again very simple. We will investigate the phenomeno-
logical consequences of an effective action that adds to the usual gluon and
quark (gauge covariant) kinetic terms the corresponding kinetic terms for
the scalars with quantum numbers of the q¯q-composites. Furthermore, this
is supplemented by a scalar potential and a Yukawa interaction between
quarks and scalars. It is very remarkable that such a simple effective action
can account for the quantum numbers, the masses and the interactions of the
pseudoscalar octet and the η mesons, the vector meson octet and the baryon
′
octet! Once the free parameters of this effective action are fixed by match-
3
ing observation, several nontrivial properties of hadrons can be “predicted”
without further unknown parameters. These findings suggest that the long-
sought dual description of long-distance strong interactions can be realized
by the addition of fields for composites. The situation would then be quite
similar to the asymptotically free nonlinear sigma model in two dimensions
where the addition of the composite “radial excitation” provides for a simple
dual description of the low momentum behavior [6].
Spontaneous breaking of color has also been proposed [7] for situations
with a very high baryon density, as perhaps in the interior of neutron stars.
In this proposal a condensation of diquark operators is responsible for color
superconductivity and spontaneous breaking of baryon number. In particu-
lar, thesuggestionofcolor-flavorlocking[8]offersanalogiestoourdescription
of the vacuum, even though different physical situations are described (va-
cuum vs. high density state) and the pattern of spontaneous color-symmetry
breaking is distinct (quark-antiquark vs. quark-quark condensate; conserved
vs. broken baryon number). This analogy may be an important key for the
understanding of possible phase transitions to a high density phase of QCD.
In the second part of this work we first present in sect. 3 our proposal
for a simple effective action for QCD. Sect. 4 describes the “Higgs picture”
with a non-vanishing expectation value of the scalar octet field. The gauge
invariantdescriptionintermsofnonlinearfieldsisintroducedinsect. 5. Sect.
6 discusses the role of a hidden local symmetry in the nonlinear description
which will permit a direct connection to ideas of “vector dominance”. The
elctromagnetic interactions can be used as an efficient probe of our picture
since no new free parameters are introduced (sect. 7). In sect. 8 we turn
to the interactions of the vector mesons and the decay ρ 2π. Here we
→
also make direct contact with the description of vector mesons as gauge
invariant q¯γµq bound states in sect. 2. Sect. 9 discusses the interactions of
the pseudoscalar mesons. We will see that the strength of the vector and
axialvector couplings of the nucleons as well as the pion interactions beyond
leading order chiral perturbation are successfully accounted for by our simple
effective action. Sect. 10 finally presents conclusions and discussion and
makes a simple proposal for the QCD phase diagram.
2 Functional integral with composite fields
Our starting point is the partition function for QCD
Z = ψ A e S (1)
−
D D
Z
4
where S is the gauge invariant classical action for quarks and gluons. Here
the quarks are described by Grassmann variables obeying ψ(x), ψ(y) =
{ }
0 , ψ(q), ψ(q ) = 0 in position and momentum space, respectively.
′
{ }
The difficult part in the functional integral (1) is the functional measure
ψ A which includes the regularization , for example on a lattice, or the
D D
gauge fixing and ghost parts for a continuum formulation. We do not need
R
a specification here and only assume that the functional measure preserves
the gauge symmetry.
It is well known that local gauge symmetries cannot be spontaneously
broken (in a strict sense). Therefore only gauge invariant quantities can have
nonzero expectation values. A prominent example concerns the correlation
functions for scalars and pseudoscalars which are contained in
¯ ¯
G (x,y) = ψ (x)ψ (x) ψ (y)ψ (y) . (2)
s L R R L
−h i
Here (ψ¯ ψ ) = (ψ¯1+γ5ψ) and(cid:0) brackets de(cid:1)n(cid:0)ote contract(cid:1)ions of spinor and
R L 2
color indices, while we have not displayed the (open) flavor indices. For the
example of the pion channel the two point function decays for large x y as
| − |
G exp( m x y ) and this is the way how the pion mass is measured,
s π
∼ − | − |
for example on the lattice. Similarly, the ρ-meson mass can be extracted
from the gauge invariant correlation function in the vector channel
Gµν(x,y) = ψ¯(x)γµψ(x) ψ¯(y)γνψ(y) . (3)
V h i
(cid:0) (cid:1)(cid:0) (cid:1)
A convenient way for the computation of correlation functions for gauge
invariant quark-antiquark bilinears is the introduction of gauge invariant
sources
S = d4x (ψ¯ (x)j (x)ψ (x)+h.c.)+ψ¯(x)γµj (x)ψ(x) . (4)
j − L s† R Vµ
Z n o
Here j is a complex N N matrix and j denotes a hermitean N N
s f f V f f
× ×
matrix, with N the number of quark flavors. We will concentrate here on
f
the three light flavors of quarks (N = 3) while considering the two flavor
f
case as a pedagogical example below. Then the physical values of the scalar
sources are given by the current quark masses
m
u
j = j = m = m (5)
s s† q  d 
m
s
 
while the vector source vanishes j = 0. Adding S to S the partition
V j
function Z[j] (1) becomes now a functional of the sources.
5
As an example, we may write the expectation value of the scalar quark-
antiquark bilinear as (a,b are flavor indices while color and spinor indices are
contracted)
δZ
ψ¯ (x)ψ (x) = σ (x) = Z 1 . (6)
Lb Ra ab −
h i δj (x)
a∗b
Similarly, the (unconnected) two point function obtains from the matrix of
the second functional variations4
G(u)(x,y) = ψ¯ (x)ψ (x) ψ¯ (y)ψ (y)
s −h R L L R i
δ2Z
= Z (cid:0)1 (cid:1).(cid:0) (cid:1) (7)
−
δj(x)δj (y)
∗
As usual, the expectation value or the connected Green’s function can be
derived from
W[j] = lnZ[j], (8)
as
δW
σ (x) = (9)
ab
δj (x)
a∗b
or
δ2W
G (x,y) = G(u)(x,y) σ (x)σ(y) = . (10)
s s − ∗ δj(x)δj (y)
∗
Finally, the effective action Γ is a functional of the “classical field” σ
Γ[σ] = W[j]+ d4x tr j (x)σ(x)+σ (x)j(x) (11)
† †
−
Z
(cid:0) (cid:1)
whereσ correspondstoagivensourcej andj = j[σ]iscomputedbyinversion
of
δW
σ[j] = . (12)
δj
∗
The field equation reads
δΓ
= j (x). (13)
δσ (x) a∗b
ab
Furthermore, thesecondfunctionalvariationofΓwithrespect toσ equalsthe
inverse connected two point function, i.e. the inverse of the second functional
variation of W with respect to j
Γ(2)W(2) = . (14)
1
4Inflavorspace(G ) isaN2 N2 matrix,withrowsandcolumnslabeledbyflavor
s ab,cd f× f
index pairs (ab) and (cd).
6
(Note that for this matrix notation all internal and space or momentum
labels of σ are collected into a vector.) Of course, this setting can be gen-
eralized to other gauge invariant composite operators, like vector fields, in a
straightforward way.
Let us concentrate here on the scalar sector and, for simplicity of the
demonstration, on N = 2. For x-independent values σ(x) = σ one has
f
Γ = d4xU(σ) where the effective potential U is now a simple function of
the complex 2x2 matrices σ. By construction U(σ) is invariant under the
R
chiral flavor rotations SU(N ) SU(N ) . If we expand U(σ) in powers of
f L f R
×
σ only invariants can appear
ν
U(σ) = m2 tr(σ σ) (detσ +detσ )
† †
− 2
2
2
λ λ 1
1 2
+ tr(σ σ) + tr σ σ tr(σ σ) +... (15)
† † †
2 2 − 2
!
(cid:18) (cid:19)
and we note that the axial U(1) -anomaly is reflected in the term ν. The
A
∼
expectation value of σ follows from the field equation
∂U
= j = m . (16)
∗ q
∂σ
This suggests to define
∂U
j
U = U tr(j σ +σ j) , = 0. (17)
j † †
− ∂σ
In particular, forequal up-and-down-quark masses the minimum of U occurs
j
typically for
σ¯
σ = (18)
h i σ¯
(cid:18) (cid:19)
and breaks the chiral SU(2) SU(2) symmetry to a vectorlike “diagonal”
L R
×
SU(2) -symmetry. Werecallthatσ¯ isdirectlyrelatedtothequark-antiquark
V
condensate
1
¯
σ¯ = ψψ . (19)
−2h i
One of the advantages of our formulation is the explicit chiral symmetry
of U(σ) independently of the quark masses, since m enters only through
q
the source term in the field equation (16). In particular, one can make
direct contact to chiral perturbation theory by representing the pseudoscalar
(pseudo-) Goldstone bosons by a nonlinear field5 U(x)
σ(x) = σ¯U(x) , U U = 1. (20)
†
5ThecoincidenceofthesymbolU forthenonlinearchiralfieldandtheeffectivepotential
is unfortunate but kept here for the sake of agreement with widely used conventions.
7
Inserting the nonlinear field (20) into the effective potential (17) yields
ν
U (U) = const m σ¯tr(U +U ) σ¯2(detU +detU ). (21)
j q † †
− − 2
In the chiral limit m = 0 the potential only involves the η meson - the
q
chiral U(1) -anomaly produces a mass term for the η-meson. In this limit
A
the pions are massless Goldstone bosons. For nonvanishing quark masses
also the pions acquire a mass m2 m .
π ∼ q
Let us neglect the η-meson and discuss explicitly the interactions of the
pions
i~τ~π(x)
detU = 1 , U(x) = exp . (22)
f
(cid:26) π (cid:27)
The kinetic term for σ results in the nonlinear kinetic term for the pions
2Z¯σ¯2
Z¯tr(∂µσ ∂ σ) Z¯σ¯2tr(∂µU ∂ U) = ∂µ~π∂ ~π +... (23)
† µ → † µ f2 µ
π
where we identify the pion decay constant
f = 2Z¯1/2σ¯. (24)
π
The effective action takes now the form familiar from chiral perturbation
theory
f2
[U] = π tr(∂µU ∂ U) 2Bm tr(U +U )+... (25)
† µ q †
L 4 −
n o
with
¯
2σ¯ ψψ
B = = h i. (26)
f2 − f2
π π
In this treatment, however, [U] is already the effective Lagrangian from
L
which the 1PI-vertices follow directly by taking suitable derivatives with
respect to the fields. No more fluctuations have to be incorporated at this
stage. In particular, there are no explicit meson fluctuations.
In order to recover chiral perturbation theory we want to reformulate our
problem such that explicit pion fluctuations are incorporated in the com-
putation of the effective action. Indeed, it is possible to reformulate the
original functional integral for QCD (1) into an exactly equivalent functional
integral which now involves explicitly an integration also over scalar fluctua-
tions. This is achieved by means of a Hubbard-Stratonovich transformation
[9]. Let us denote the scalar fermion bilinears by
¯
σ˜ = ψ ψ , σ˜ = σ . (27)
ab Lb Ra ab ab
h i
8
We next introduce a unit into the functional integral (1)
d4q
= N σ exp tr (σ λ σ˜ j) λ 1(σ λ σ˜ j)
1 D ′ − (2π)4 ′ − σ − † −σ ′ − σ −
Z (cid:26) Z (cid:27)
n o (28)
where N is an irrelevant normalization constant. The functional integral for
the partition function Z[j] involves now an additional integration over the
scalar field σ
′
(σ)
Z[j] = ψ A σ exp( S ). (29)
D D D ′ − j
Z
(σ)
Here the action S in eq. (4) is replaced by S , i.e.
j j
d4q
S(σ) = S + tr σ λ 1σ (σ σ˜ +σ˜ σ )+λ σ˜ σ˜
j 0 (2π)4 ′† −σ ′ − ′† † ′ σ †
Z n
(j λ 1σ +σ λ 1j)+j λ 1j (30)
− † −σ ′ ′† −σ † −σ
o
where S is the QCD-action for quarks and gluons without the quark mass
0
term. Indeed, the quark mass term tr(j σ) is canceled by a corresponding
†
∼
term from eq. (28) such that the nonvanishing quark masses appear now as
a source term multiplying the scalar field σ .
e′
Ineq. (28)σ standsforσ (q)andλ 1(q)isanarbitrarypositive function
′ a′b −σ
of q2 such that the Gaussian integral is well defined and rotation symmetry
preserved. If we choose
1
λ = (31)
σ
M2 +Zq2
we can identify the term σ λ 1σ with a kinetic and mass term for σ such
′† −σ ′ ′
that λ corresponds to the “classical scalar propagator”. The term σ˜ σ de-
σ † ′
notes a Yukawa coupling of the σ field to the quarks. Finally, the expression
′
λ σ˜ σ˜ denotes a four quark interaction that is nonlocal for Z = 0.
σ †
6
The explicit four-quark-interaction is cumbersome and we may wish to
omit it. This can be done if we introduce into the original action for quarks
and gluons an additional four-quark-interaction with the opposite sign - this
is then canceled by the piece arising from the Hubbard-Stratonovich trans-
formation. At first sight, this seems to be a high prize to pay since we are not
dealing any more with the standard QCD action where the quarks interact
only via gluon exchange. With a second look, however, this is no problem.
Implementing this modified QCD action for lattice simulations will even re-
sult in an “improved” QCD action if suitable values for M2 and Z are chosen
in eq. (31).
The basic ingredient for this argument is universality. In QCD the pre-
cise form of the short distance action (classical action) is actually irrelevant,
9